   Chapter 3.2, Problem 24E

Chapter
Section
Textbook Problem

# (a) Suppose that f is differentiable on ℝ and has two roots. Show that f′ has at least one root.(b) Suppose / is twice differentiable on ℝ and has three roots. Show that f′′ has at least one real root.(c) Can you generalize parts (a) and (b)?

To determine

(a)

To show:

f has at least one real root

Explanation

1) Concept:

Using the Rolle’s Theorem verify the result

2) Theorem:

Rolle’s theorem- Let f be the function that satisfies the following hypotheses:

1) f is continuous on closed interval [a,b]

2) f is differentiable on open interval (a,b)

3) fa=fb

Then, there is number c in (a,b) such that f'c=0

[ if a and b are roots of l f(x), number c  is root of f(x) means between two real roots of f(x), there is one real root of f(x)]

3) Given:

f is differentiable on R and has two real roots

3) Calculation:

f is differentiable on R has two roots a and b.

fx is continuous on R

f has two roots

By Rolle’s Theorem, there is a number

c  with a<c<b

And f'c=0

Therefore, the function f'(x) has at least one real root.

Conclusion:

Therefore, f has at least one real root.

(b)

To show:

f  has at least one real root

Solution:

f  has at least one real root

1) Concept:

Using the Rolle’s Theorem verify the result

2) Theorem:

a) Roll’s theorem- Let f be function that satisfies the following hypothesis:

1) f is continuous on closed interval [a,b]

2) f is differentiable on open interval (a,b)

3) fa=fb

Then, there is number c in (a,b) such that f'c=0

[if a  and b are roots of a given function f(x), number c is root of f(x)

means between two real roots of f(x), there is one real root of f(x)]

3) Given:

f is twice differentiable on R has three roots

4) Calculation:

f is twice differentiable on R has three roots

Suppose f has a<b< c   roots

Therefore,

f'a=f'

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